By E. F. Brezzi

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All ll3-configurations and all 12a-configurations (in the classiflcation of R. Daublebsky) are realizable with rational coordinates. Here is another problem of Gr/inbaum in the same spirit. 10. Is there a reM realizable n4-conflguration t'or some n <_ 20 ? 42 [123], [1501, [189], [257], [2681, [3691, [370], [456], [4781, [490] 2 z , := ( 1 , 0 , 0 ) T, zs := (0,1,0) T, 2 x9 := (0,0,1) T, 2 2 (i) (2) xo := ( 1 , - 1 , 1 ) T x~ := a6x7 + fl6xs + ~'6x9 (3) x5 := o~5x4 + ¢~5x~ x4 : = [7891xo + [8701x9 x3 := [6971xo + [960]x7 • 2 := [576]x8 + [758]x6 • ~ := a5 := 3, f15 := 1, ({3 12 4 6 -9 [5o8]~ + [o59]~ a6 := - 4 , 1 -I-4 -1 1 -1 0 -2 3 f16 := 1, 9'6 := 3 1 0 0 1) 1 3 0 0 5 / 1 0 0 1 -1 1 4 0 I 2 (4) 6 3 F i g u r e 3-4.

The (stretchable) arrangement adjacent triangles. A~2 of 12 pseudo-lines without { 1 , 2 , . . , 9 , 0 , A , B } 3 ~ { - 1 , + 1 } such that X12 is realizable if and only if ,4~2 is stretchable. 12, we determine a reduced system 74 for X12 which is listed in Table 3-1. The system 74 contains 46 of the (½2) = 220 bases. Recall that, by the results of [131], the triangles of Jt~2 are necessarily contained in every reduced system 74 of X12- 54 [124]+ [13A1+ [23A1+ [240] + [2781~-B-]--+ [125]+ [245][24A] + [27A]+ [34A1+ [126]+ [14B1+ [2461[24B] + [28A]+ [359]- [127]+ [16A][247][25A] + [29A]+ [37A]- [128]+ [1701+ [2481[269] + ~ [38B1+ [45A]- [468]- [46A~- [47A]- ~ [40A]+ [4AB]- [5671+ [5AB]+ [80A]- [14A]- [12A]+ [234][249][26A] + [20BI- Table 3-1.

Realizations over Q are motivated by the "diophantine problems" that we mentioned in the introduction. Here we prove that for these two fields the realizability of matroids and oriented matroids and the Steinitz problem are equivalent to solving a single polynomial equation. 7. Let K denote either the rationals Q or its reM closure, the reM algebraic numbers A . Then the following statements are equivalent. (1) There exists an algorithm for deciding an arbitrary polynomial f 6 Z [ x l , . . , x , ] , n 6 N, whether f has zeros in K n.